Publications of John Martin
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See detailChaotic Bohmian trajectories for stationary states
Cesa, Alexandre ULg; Martin, John ULg; Struyve, Ward

in Journal of Physics A: Mathematical and Theoretical (2016), 49

In Bohmian mechanics, the nodes of the wave function play an important role in the generation of chaos. However, so far, most of the attention has been on moving nodes; little is known about the ... [more ▼]

In Bohmian mechanics, the nodes of the wave function play an important role in the generation of chaos. However, so far, most of the attention has been on moving nodes; little is known about the possibility of chaos in the case of stationary nodes. We address this question by considering stationary states, which provide the simplest examples of wave functions with stationary nodes. We provide examples of stationary wave functions for which there is chaos, as demonstrated by numerical computations, for one particle moving in 3 spatial dimensions and for two and three entangled particles in two dimensions. Our conclusion is that the motion of the nodes is not necessary for the generation of chaos. What is important is the overall complexity of the wave function. That is, if the wave function, or rather its phase, has complex spatial variations, it will lead to complex Bohmian trajectories and hence to chaos. Another aspect of our work concerns the average Lyapunov exponent, which quantifies the overall amount of chaos. Since it is very hard to evaluate the average Lyapunov exponent analytically, which is often computed numerically, it is useful to have simple quantities that agree well with the average Lyapunov exponent. We investigate possible correlations with quantities such as the participation ratio and different measures of entanglement, for different systems and different families of stationary wave functions. We find that these quantities often tend to correlate to the amount of chaos. However, the correlation is not perfect, because, in particular, these measures do not depend on the form of the basis states used to expand the wave function, while the amount of chaos does. [less ▲]

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See detailCooperative spontaneous emission from indistinguishable atoms in arbitrary motional quantum states
Damanet, François ULg; Braun, Daniel; Martin, John ULg

in Physical Review A (2016), 94

We investigate superradiance and subradiance of indistinguishable atoms with quantized motional states, starting with an initial total state that factorizes over the internal and external degrees of ... [more ▼]

We investigate superradiance and subradiance of indistinguishable atoms with quantized motional states, starting with an initial total state that factorizes over the internal and external degrees of freedom of the atoms. Due to the permutational symmetry of the motional state, the cooperative spontaneous emission, governed by a recently derived master equation [F. Damanet et al., Phys. Rev. A 93, 022124 (2016)], depends only on two decay rates γ and γ0 and a single parameter dd describing the dipole-dipole shifts. We solve the dynamics exactly for N = 2 atoms, numerically for up to 30 atoms, and obtain the large-N limit by a mean-field approach. We find that there is a critical difference γ0 − γ that depends on N beyond which superradiance is lost. We show that exact nontrivial dark states (i.e., states other than the ground state with vanishing spontaneous emission) only exist for γ = γ0 and that those states (dark when γ = γ0) are subradiant when γ < γ0. [less ▲]

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See detailMaster equation with quantized atomic motion including dipole-dipole interactions
Damanet, François ULg; Braun, Daniel; Martin, John ULg

Scientific conference (2016, May 26)

We derive a markovian master equation for the internal dynamics of an ensemble of two-level atoms including all effects related to the quantization of their motion [1]. Our equation provides a unifying ... [more ▼]

We derive a markovian master equation for the internal dynamics of an ensemble of two-level atoms including all effects related to the quantization of their motion [1]. Our equation provides a unifying picture of the consequences of recoil and in- distinguishability of atoms beyond the Lamb-Dicke regime on both their dissipative and conservative dynamics, and is relevant for experiments with ultracold trapped atoms. We give general expressions for the decay rates and the dipole-dipole shifts for any motional states, and we find analytical formulas for a number of relevant states (Gaussian states, Fock states and thermal states). In particular, we show that the dipole-dipole interactions and cooperative photon emission can be modulated through the external state of motion. The effects predicted should be experimen- tally observable with Rydberg atoms [2]. [1] F. Damanet, D. Braun, and J. Martin, arXiv:1512.06676v2. [2] K. Afrousheh, P. Bohlouli-Zanjani, D. Vagale, A. Mugford, M. Fedorov, and J. D. D. Martin, Phys. Rev. Lett. 93, 233001 (2004) [less ▲]

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See detailChaotic Bohmian trajectories for stationary states
Cesa, Alexandre ULg; Martin, John ULg; Struyve, Ward

Poster (2016, May 18)

We study the possibility of chaos for the Bohmian dynamics when the wave function is stationary. Examples of stationary wave functions are given for which there is chaos, as demonstrated by numerical ... [more ▼]

We study the possibility of chaos for the Bohmian dynamics when the wave function is stationary. Examples of stationary wave functions are given for which there is chaos, as demonstrated by numerical computations, for one particle moving in 3 spatial dimensions and for two and three entangled particles in 2 dimensions. What is important for the amount of chaos is the overall complexity of the wave function. Some simple measures that partly capture the complexity of the wave function are considered: the participation ratio and different measures of entanglement. We find that these measures often tend to correlate to the amount of chaos. However, the correlation is not perfect, because the measures do not depend on the intrinsic complexity of the states of a given basis. [less ▲]

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See detailSuper- and subradiance from indistinguishable atoms with quantized motion
Damanet, François ULg; Braun, Daniel; Martin, John ULg

Poster (2016, May 18)

Detailed reference viewed: 23 (4 ULg)
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See detailAnticoherence of spin states with point-group symmetries
Baguette, Dorian ULg; Damanet, François ULg; Giraud, Olivier et al

Poster (2016, May 18)

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See detailCollective spontaneous emission with quantized atomic motion
Damanet, François ULg; Braun, Daniel; Martin, John ULg

Scientific conference (2016, May 09)

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See detailScattering theory of walking droplets in the presence of obstacles
Dubertrand, Rémy ULg; Hubert, Maxime ULg; Schlagheck, Peter ULg et al

E-print/Working paper (2016)

We aim to describe a droplet bouncing on a vibrating bath. Due to Faraday instability a surface wave is created at each bounce and serves as a pilot wave of the droplet. This leads to so called walking ... [more ▼]

We aim to describe a droplet bouncing on a vibrating bath. Due to Faraday instability a surface wave is created at each bounce and serves as a pilot wave of the droplet. This leads to so called walking droplets or walkers. Since the seminal experiment by {\it Couder et al} [Phys. Rev. Lett. {\bf 97}, 154101 (2006)] there have been many attempts to accurately reproduce the experimental results. Here we present a simple and highly versatile model inspired from quantum mechanics. We propose to describe the trajectories of a walker using a Green function approach. The Green function is related to Helmholtz equation with Neumann boundary conditions on the obstacle(s) and outgoing conditions at infinity. For a single slit geometry our model is exactly solvable and reproduces some general features observed experimentally. It stands for a promising candidate to account for the presence of any boundaries in the walkers'dynamics. [less ▲]

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See detailMaster equation for collective spontaneous emission with quantized atomic motion
Damanet, François ULg; Braun, Daniel; Martin, John ULg

Scientific conference (2016, March 02)

We derive a markovian master equation for the internal dynamics of an ensemble of two-level atoms including the quantization of their motion. Our equation provides a unifying picture of the effects of ... [more ▼]

We derive a markovian master equation for the internal dynamics of an ensemble of two-level atoms including the quantization of their motion. Our equation provides a unifying picture of the effects of recoil and indistinguishability of atoms beyond the Lamb-Dicke regime on both their dissipative and conservative dynamics. We give general expressions for the decay rates and the dipole-dipole shifts for any motional states, generalizing those in Ref. [1]. We find closed-form formulas for a number of relevant states (gaussian states, Fock states and thermal states). In particular, we show that dipole-dipole interactions and cooperative photon emission [2] can be modulated through the external state of motion. As an application of our general formalism, we study the spatial Pauli blocking of two fermionic atoms beyond the Lamb-Dicke regime [3]. [1] G. S. Agarwal, Springer Tracts In Modern Physics 70, 1 (1974). [2] R. H. Dicke, Phys. Rev. 93, 99 (1954). [3] R. M. Sandner, M. Müller, A. J. Daley & P. Zoller, Phys. Rev. A 84, 043825 (2011). [less ▲]

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See detailAnticoherence of spin states with point-group symmetries
Baguette, Dorian ULg; Damanet, François ULg; Giraud, Olivier et al

Poster (2016, March 02)

We investigate multiqubit permutation-symmetric states with maximal entropy of entanglement. Such states can be viewed as particular spin states, namely anticoherent spin states. Using the Majorana ... [more ▼]

We investigate multiqubit permutation-symmetric states with maximal entropy of entanglement. Such states can be viewed as particular spin states, namely anticoherent spin states. Using the Majorana represen- tation of spin states in terms of points on the unit sphere, we analyze the consequences of a point-group symmetry in their arrangement on the quantum properties of the corresponding state [1]. We focus on the identi cation of anticoherent states (for which all reduced density matrices in the symmetric subspace are maximally mixed) associated with point-group-symmetric sets of points. We provide three di erent characterizations of anticoherence and establish a link between point symmetries, anticoherence, and classes of states equivalent through stochastic local operations with classical communication. We then in- vestigate in detail the case of small numbers of qubits and construct in nite families of anticoherent states with point-group symmetry of their Majorana points, showing that anticoherent states do exist to arbitrary order. [1] D. Baguette et al., Phys. Rev. A 92, 052333 (2015). [less ▲]

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See detailAnticoherence and entanglement of spin states
Martin, John ULg; Baguette, Dorian ULg; Damanet, François ULg et al

Scientific conference (2016, March 02)

We investigate multiqubit permutation-symmetric states with maxi- mally mixed reduced density matrices in the symmetric subspace [1]. Such states can be viewed as particular spin states, namely anticoher ... [more ▼]

We investigate multiqubit permutation-symmetric states with maxi- mally mixed reduced density matrices in the symmetric subspace [1]. Such states can be viewed as particular spin states, namely anticoher- ent spin states [2]. Using the Majorana representation of spin states in terms of points on the unit sphere [3], we analyze the consequences of degeneracies of the Majorana points and of a point-group symmetry in their arrangement on the existence of anticoherent spin states. We provide different characterizations of anticoherence and establish a link between point symmetries, anticoherence, and SLOCC classes [4]. We consider in detail the case of small numbers of qubits and solve the 4-qubit case completely by identifying and characterizing all 4-qubit anticoherent states. [1] D. Baguette, T. Bastin, and J. Martin, Phys. Rev. A 90, 032314 (2014); O. Giraud et al., Phys. Rev. Lett. 114, 080401 (2015); D. Baguette et al., Phys. Rev. A 92, 052333 (2015). [2] J. Zimba, Electron. J. Theor. Phys. 3, 143 (2006). [3] E. Majorana, Nuovo Cimento 9, 43 (1932). [4] SLOCC classes : Classes of states equivalent through stochastic local operations with classical communication. [less ▲]

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See detailMaster equation for collective spontaneous emission with quantized atomic motion
Damanet, François ULg; Braun, Daniel; Martin, John ULg

in Physical Review A (2016), 93

We derive a Markovian master equation for the internal dynamics of an ensemble of two-level atoms including all effects related to the quantization of their motion. Our equation provides a unifying ... [more ▼]

We derive a Markovian master equation for the internal dynamics of an ensemble of two-level atoms including all effects related to the quantization of their motion. Our equation provides a unifying picture of the consequences of recoil and indistinguishability of atoms beyond the Lamb-Dicke regime on both their dissipative and conservative dynamics, and applies equally well to distinguishable and indistinguishable atoms. We give general expressions for the decay rates and the dipole-dipole shifts for any motional states, and we find closed-form formulas for a number of relevant states (Gaussian states, Fock states, and thermal states). In particular, we show that dipole-dipole interactions and cooperative photon emission can be modulated through the external state of motion. [less ▲]

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See detailAnticoherence of spin states with point-group symmetries
Baguette, Dorian ULg; Damanet, François ULg; Giraud, Olivier et al

in Physical Review A (2015), 92

We investigate multiqubit permutation-symmetric states with maximal entropy of entanglement. Such states can be viewed as particular spin states, namely anticoherent spin states. Using the Majorana ... [more ▼]

We investigate multiqubit permutation-symmetric states with maximal entropy of entanglement. Such states can be viewed as particular spin states, namely anticoherent spin states. Using the Majorana representation of spin states in terms of points on the unit sphere, we analyze the consequences of a point-group symmetry in their arrangement on the quantum properties of the corresponding state. We focus on the identification of anticoherent states (for which all reduced density matrices in the symmetric subspace are maximally mixed) associated with point-group-symmetric sets of points. We provide three different characterizations of anticoherence and establish a link between point symmetries, anticoherence, and classes of states equivalent through stochastic local operations with classical communication. We then investigate in detail the case of small numbers of qubits and construct infinite families of anticoherent states with point-group symmetry of their Majorana points, showing that anticoherent states do exist to arbitrary order. [less ▲]

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See detailMultifractality of quantum wave functions in the presence of perturbations
Dubertrand, Rémy ULg; Garcia-Mata, Ignacio; Georgeot, Bertrand et al

in Physical Review. E : Statistical, Nonlinear, and Soft Matter Physics (2015), 92

We present a comprehensive study of the destruction of quantum multifractality in the presence of perturbations. We study diverse representative models displaying multifractality, including a ... [more ▼]

We present a comprehensive study of the destruction of quantum multifractality in the presence of perturbations. We study diverse representative models displaying multifractality, including a pseudointegrable system, the Anderson model, and a random matrix model. We apply several types of natural perturbations which can be relevant for experimental implementations. We construct an analytical theory for certain cases and perform extensive large-scale numerical simulations in other cases. The data are analyzed through refined methods including double scaling analysis. Our results confirm the recent conjecture that multifractality breaks down following two scenarios. In the first one, multifractality is preserved unchanged below a certain characteristic length which decreases with perturbation strength. In the second one, multifractality is affected at all scales and disappears uniformly for a strong-enough perturbation. Our refined analysis shows that subtle variants of these scenarios can be present in certain cases. This study could guide experimental implementations in order to observe quantum multifractality in real systems. [less ▲]

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See detailChaotic Bohmian trajectories for the hydrogen atom
Cesa, Alexandre ULg; Struyve, Ward ULg; Martin, John ULg

Poster (2015, May 13)

In Bohmian mechanics, a single-particle quantum system is described in part by its wave function and in part by the actual position of the particle. The trajectory of the latter can be computed using the ... [more ▼]

In Bohmian mechanics, a single-particle quantum system is described in part by its wave function and in part by the actual position of the particle. The trajectory of the latter can be computed using the guiding equation. This equation states that the velocity of the particle is proportional to the usual probability current associated with its wave function. In this work, we study the quantum trajectory of a single particle in a Coulomb potential whose eigenstates are the well known eigenstates of the hydrogen atom. More precisely, we focus on the relation between chaotic Bohmian trajectories and the motion of wave function nodes. At wave function nodes i.e., where the wave function vanishes, the velocity is not defined which generically induces vorticity. In order to probe chaos, we compute Poincaré map and we numerically evaluate Lyapounov exponents, which characterize the divergence of close trajectories as time increases. For the 2d Coulomb potential, although the superposition of two eigenstates with different energies can lead to an arbitrary high number of moving nodes of the wave function, the Bohmian trajectories display no trace of chaos. This absence of chaotic behaviour originates from the existence of a constant of motion. Therefore, the motion and the number of nodes do not constitute a sufficient condition for the emergence of chaos in Bohmian mechanics. For superpositions of more than two eigenstates, there is no constant of motion, there are moving nodes and we find that the Bohmian trajectories are chaotic. [less ▲]

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See detailAnticoherence of multiqubit symmetric states
Baguette, Dorian ULg; Bastin, Thierry ULg; Martin, John ULg

Conference (2015, May 13)

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See detailTensor Representation of Spin States
Giraud, Olivier; Braun, Daniel; Baguette, Dorian ULg et al

in Physical Review Letters (2015), 114

We propose a generalization of the Bloch sphere representation for arbitrary spin states. It provides a compact and elegant representation of spin density matrices in terms of tensors that share the most ... [more ▼]

We propose a generalization of the Bloch sphere representation for arbitrary spin states. It provides a compact and elegant representation of spin density matrices in terms of tensors that share the most important properties of Bloch vectors. Our representation, based on covariant matrices introduced by Weinberg in the context of quantum field theory, allows for a simple parametrization of coherent spin states, and a straightforward transformation of density matrices under local unitary and partial tracing operations. It enables us to provide a criterion for anticoherence, relevant in a broader context such as quantum polarization of light. [less ▲]

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See detailMultiqubit symmetric states with maximally mixed one-qubit reductions
Baguette, Dorian ULg; Bastin, Thierry ULg; Martin, John ULg

Poster (2014, November 18)

We present a comprehensive study on the remarquable properties shared by maximally entangled symmetric states of arbitrary numbers of qubits in the sense of the maximal mixedness of the one-qubit reduced ... [more ▼]

We present a comprehensive study on the remarquable properties shared by maximally entangled symmetric states of arbitrary numbers of qubits in the sense of the maximal mixedness of the one-qubit reduced density operator. Such states are of great interest in quantum information as they maximize several measures of entanglement, such as Meyer-Wallach entropy [1] and any entanglement monotone based on linear homogenous positive functions of pure state within their SLOCC classes of states [2, 3]. When they exist, they are unique up to local unitaries within their SLOCC classes [3, 4]. They play a specific role in the determination of the local unitary equivalence of multiqubit states [5]. Moreover, they are maximally fragile (in the sense that they are the states which are the most sensitive to noise) and have therefore been proposed as ideal candidates for ultrasensitive sensors [6]. They appear in the litterature under various names : maximally entangled states [6], 1-uniform states [7], normal forms [3, 4] and nongeneric states [5]. We present a general criterion to easily identify whether given symmetric states are maximally entangled or not [9]. We show that these maximally entangled symmetric (MES) states are the only symmetric states for which the expectation value of the associated collective spin S of the system vanishes, which coincides with the definition of anticoherence to order one of spin states. This definition also coincides with the cancellation of the dipole moment of the Husimi function of the state. We then generalize these properties and show that a state is anticoherent to order t, <(S.n)^k> is independent of n for k = 1, . . . , t, where n is a unit vector, iff it has maximally mixed t-qubit reductions or iff all moments up to order 2t of its Husimi function vanish. We also establish the equivalence between anticoherent states to order t and unpolarized light states to order t [8], thereby encompassing various state characterizations under the same banner [9, 10]. We provide a nonexistence criterion allowing us to know immediately whether SLOCC classes of symmetric states can contain MES states or not. We show in particular that the symmetric Dicke state SLOCC classes never contain such MES states, with the only exception of the balanced Dicke state class for even numbers of qubits. We analyze the 4-qubit system exhaustively and identify and characterize all MES states of this system as well as the only 4-qubit state anticoherent to order 2. Finally, we analyze the entanglement content of MES states with respect to the geometric [11] and barycentric [12] measures of entanglement. [1] D. A. Meyer and N. R. Wallach, J. Math. Phys. 43, 4273 (2002). [2] Classes of states equivalent through stochastic local operations with classical communication. [3] F. Verstraete, J. Dehaene, and B. De Moor, Phys. Rev. A 68, 012103 (2003). [4] G. Gour and N. Wallach, N. J. Phys. 13, 073013 (2011). [5] B. Kraus, Phys. Rev. Lett. 104, 020504 (2010). [6] N. Gisin and H. Bechmann-Pasquinucci, Phys. Lett. A 246, 1 (1998). [7] A. J. Scott, Phys. Rev. A 69, 052330 (2004). [8] L. L. Sánchez-Soto, A. B. Klimov, P. de la Hoz, and G. Leuchs J. Phys. B : At. Mol. Opt. Phys. 46, 104011 (2013). [9] D. Baguette, T. Bastin, and J. Martin, Phys. Rev. A 90, 032314 (2014). [10] O. Giraud, D. Braun, D. Baguette, T. Bastin, and J. Martin, arXiv :1409.1106. [11] T.-C. Wei and P. M. Goldbart, Phys. Rev. A 68, 042307 (2003). [12] W. Ganczarek, M. Kus, and K. Zyczkowski, Phys. Rev. A 85, 032314 (2012). [less ▲]

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See detailOn the analogy of quantum wave-particle duality with bouncing droplets
Richardson, Chris D.; Schlagheck, Peter ULg; Martin, John ULg et al

E-print/Working paper (2014)

We explore the hydrodynamic analogues of quantum wave-particle duality in the context of a bouncing droplet system which we model in such a way as to promote comparisons to the de Broglie-Bohm ... [more ▼]

We explore the hydrodynamic analogues of quantum wave-particle duality in the context of a bouncing droplet system which we model in such a way as to promote comparisons to the de Broglie-Bohm interpretation of quantum mechanics. Through numerical means we obtain single-slit diffraction and double-slit interference patterns that strongly resemble those reported in experiment and that reflect a striking resemblance to quantum diffraction and interference on a phenomenological level. We, however, identify evident differences from quantum mechanics which arise from the governing equations at the fundamental level. [less ▲]

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See detailMultiqubit symmetric states with maximally mixed one-qubit reductions
Baguette, Dorian ULg; Bastin, Thierry ULg; Martin, John ULg

in Physical Review A (2014), 90

We present a comprehensive study of maximally entangled symmetric states of arbitrary numbers of qubits in the sense of the maximal mixedness of the one-qubit reduced density operator. A general criterion ... [more ▼]

We present a comprehensive study of maximally entangled symmetric states of arbitrary numbers of qubits in the sense of the maximal mixedness of the one-qubit reduced density operator. A general criterion is provided to easily identify whether given symmetric states are maximally entangled in that respect or not. We show that these maximally entangled symmetric (MES) states are the only symmetric states for which the expectation value of the associated collective spin of the system vanishes, as well as in corollary the dipole moment of the Husimi function. We establish the link between this kind of maximal entanglement, the anticoherence properties of spin states, and the degree of polarization of light fields. We analyze the relationship between the MES states and the classes of states equivalent through stochastic local operations with classical communication (SLOCC). We provide a nonexistence criterion of MES states within SLOCC classes of qubit states and show in particular that the symmetric Dicke state SLOCC classes never contain such MES states, with the only exception of the balanced Dicke state class for even numbers of qubits. The 4-qubit system is analyzed exhaustively and all MES states of this system are identified and characterized. Finally the entanglement content of MES states is analyzed with respect to the geometric and barycentric measures of entanglement, as well as to the generalized N-tangle. We show that the geometric entanglement of MES states is ensured to be larger than or equal to 1/2, but also that MES states are not in general the symmetric states that maximize the investigated entanglement measures. [less ▲]

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